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 A191724 Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals. 20

%I

%S 1,3,2,8,5,4,20,13,10,6,50,33,25,15,7,125,83,63,38,18,9,313,208,158,

%T 95,45,23,11,783,520,395,238,113,58,28,12,1958,1300,988,595,283,145,

%U 70,30,14,4895,3250,2470,1488,708,363,175,75,35,16,12238,8125,6175

%N Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.

%C For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.

%C ...

%C Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:

%C ...

%C A191722=dispersion of A008851 (0, 1 mod 5 and >1)

%C A191723=dispersion of A047215 (0, 2 mod 5 and >1)

%C A191724=dispersion of A047218 (0, 3 mod 5 and >1)

%C A191725=dispersion of A047208 (0, 4 mod 5 and >1)

%C A191726=dispersion of A047216 (1, 2 mod 5 and >1)

%C A191727=dispersion of A047219 (1, 3 mod 5 and >1)

%C A191728=dispersion of A047209 (1, 4 mod 5 and >1)

%C A191729=dispersion of A047221 (2, 3 mod 5 and >1)

%C A191730=dispersion of A047211 (2, 4 mod 5 and >1)

%C A191731=dispersion of A047204 (3, 4 mod 5 and >1)

%C ...

%C A191732=dispersion of A047202 (2,3,4 mod 5 and >1)

%C A191733=dispersion of A047206 (1,3,4 mod 5 and >1)

%C A191734=dispersion of A032793 (1,2,4 mod 5 and >1)

%C A191735=dispersion of A047223 (1,2,3 mod 5 and >1)

%C A191736=dispersion of A047205 (0,3,4 mod 5 and >1)

%C A191737=dispersion of A047212 (0,2,4 mod 5 and >1)

%C A191738=dispersion of A047222 (0,2,3 mod 5 and >1)

%C A191739=dispersion of A008854 (0,1,4 mod 5 and >1)

%C A191740=dispersion of A047220 (0,1,3 mod 5 and >1)

%C A191741=dispersion of A047217 (0,1,2 mod 5 and >1)

%C ...

%C For further information about these 20 dispersions, see A191722.

%C ...

%C Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

%H Ivan Neretin, <a href="/A191724/b191724.txt">Table of n, a(n) for n = 1..5050</a>

%e Northwest corner:

%e 1....3....8....20....50

%e 2....5....13...33....83

%e 4....10...25...63....158

%e 6....15...38...95....238

%e 7....18...45...113...283

%e 9....23...58...145...363

%t (* Program generates the dispersion array t of the increasing sequence f[n] *)

%t r = 40; r1 = 12; c = 40; c1 = 12;

%t a=3; b=5; m[n_]:=If[Mod[n,2]==0,1,0];

%t f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]

%t Table[f[n], {n, 1, 30}] (* A047218 *)

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

%t rows = {NestList[f, 1, c]};

%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

%t t[i_, j_] := rows[[i, j]];

%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191724 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191724 *)

%Y Cf. A032793, A047218, A191734, A191722, A191426.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 13 2011

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Last modified April 7 19:24 EDT 2020. Contains 333306 sequences. (Running on oeis4.)